What is an ANCOVA?

1. One-way Analysis of Covariance
(ANCOVA)
Conceptual Tutorial

2. First
How did we get here?

3. Consider a similar problem

4. A pizza café owner wants to know which type of high
school athlete she should market to.

5. A pizza café owner wants to know which type of high
school athlete she should market to. Should she market
to football, basketball or soccer players?

6. A pizza café owner wants to know which type of high
school athlete she should market to. Should she market
to football, basketball or soccer players?
So she measures the ounces of pizza eaten by 12 football,
12 basketball, and 12 soccer players in one sitting.

7. Here are the raw data:
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

8. The owner wondered how much these athletes like pizza
to begin with and how that might affect the results.

9. The owner wondered how much these athletes like pizza
to begin with and how that might affect the results. She
surveyed them prior to their eating the pizza.

10. The Survey
On a scale of 1 to 10 how
much do you like pizza?
1 2 3 4 5 6 7 8 9 10

11. Here were the results:
Football Basketball Soccer
7.0 3.0 7.5
5.0 8.0 4.5
3.5 4.5 3.5
9.0 9.5 6.0
7.0 6.5 6.0
8.0 7.0 4.5
6.5 7.5 6.0
7.5 9.0 1.5
2.5 8.5 6.5
9.0 4.0 5.0
8.0 7.5 5.5
5.0 8.0 4.0

12. Based on this information, let’s determine how we got
here.

13. Here is the problem again:
A pizza café owner wants to know which type of
high school athlete she should market to, by
comparing how many ounces of pizza are
consumed across all three athlete groups.
She will control for pizza preference.

14. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.

15. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is This:
Inferential or Descriptive

16. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is This:
Inferential or Descriptive

17. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is This:
Inferential or Descriptive
Based on the data set of 36 athletes, this is a
sample from which the owner would like to make
generalizations about potential athlete customers.

18. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is This:
Inferential or Descriptive
Based on the data set of 36 athletes, this is a
sample from which the owner would like to make
generalizations about potential athlete customers.

19. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is This Question of:
Relationship or Difference

20. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is This Question of:
Relationship or Difference
Because the owner wants to compare groups differences,
we are dealing with DIFFERENCE.

21. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is This Question of:
Relationship or Difference
Because the owner wants to compare groups differences,
we are dealing with DIFFERENCE.

22. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential

23. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is The Distribution:
Normal or Not Normal

24. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is The Distribution:
Normal or Not Normal
After graphing each
column we find that the
distributions are mostly
normal.
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

25. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is The Distribution:
Normal or Not Normal
After graphing each
column we find that the
distributions are mostly
normal.
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

26. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential
Normal Not Normal

27. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Are the Data:
Scaled?
(ratio/interval/ordinal)
Categorical?
(ordinal)
or

28. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Are the Data:
Scaled?
(ratio/interval/ordinal)
Categorical?
(ordinal)
or
The data is interval (ounces of
Pizza)
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

29. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Are the Data:
Scaled?
(ratio/interval/ordinal)
Categorical?
(ordinal)
or
The data is interval (ounces of
Pizza)
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

30. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential
Normal Not Normal
Scaled Categorical

31. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 DV or 2 or more DV

32. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 DV or 2 or more DV

33. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 DV or 2 or more DV

34. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV

35. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 IV or 2 or more IVs
[Type of Athlete is the only Independent Variable (IV)]

36. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 IV or 2 or more IVs
[Type of Athlete is the only Independent Variable (IV)]

37. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 IV or 2 or more IVs
[Type of Athlete is the only Independent Variable (IV)]

38. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
1 IV 2 or more IV

39. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 IV Level
2 or more IV
Levels or

40. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 IV Level
2 or more IV
Levels or

41. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is there:
1 IV Level
2 or more IV
Levels or

42. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential
Normal Not Normal
Scaled Categorical
1 DV
2 or more
DV
1 IV 2 or more IV
2 or more IV
Levels
1 IV Level

43. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Are the Samples:
Repeated or Independent

44. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Are the Samples:
Repeated or Independent
No individual is in more than one group

45. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Are the Samples:
Repeated or Independent
No individual is in more than one group

46. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
1 IV 2 or more IV
2 or more IV
Levels
1 IV Level
Repeated Independent

47. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is There:
A Covariate or Not a Covariate

48. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is There:
A Covariate or Not a Covariate

49. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Is There:
A Covariate or Not a Covariate

50. The Problem: A pizza café owner wants to know which type of high school athlete she
should market to, by comparing how many ounces of pizza are consumed across all three
athlete groups. She will control for pizza preference.
Inferential Descriptive
Descriptive Inferential
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
1 IV 2 or more IV
1 IV Level 2 or more IV Levels
Repeated Independent
A Covariate Not a Covariate

51. Now that we know how we got here, let’s consider what
Analysis of Covariance is.

52. Now that we know how we got here, let’s consider what
Analysis of Covariance is.
First, . . . what is covariance?

53. Now that we know how we got here, let’s consider what
Analysis of Covariance is.
First, . . . what is covariance?
As you know, variance is a statistic that helps you
determine how much the data in a distribution varies.

54. Now that we know how we got here, let’s consider what
Analysis of Covariance is.
First, . . . what is covariance?
As you know, variance is a statistic that helps you
determine how much the data in a distribution varies.
5 6 7
Number of
Pizza Slices
eaten by
Basketball
Players
Not much
variance

55. Now that we know how we got here, let’s consider what
Analysis of Covariance is.
First, . . . what is covariance?
As you know, variance is a statistic that helps you
determine how much the data in a distribution varies.
5 6 7
Number of
Pizza Slices
eaten by
Basketball
Players
Not much
variance
Number of
Pizza Slices
eaten by
Soccer Players
2 3 4 5 6 7 8 9 10
A lot of
variance

56. Covariance is a statistic that helps us determine how
much two distributions that have some relationship
covary.

57. Let’s imagine that students take a math test and their
ordered scores look like this.

58. Let’s imagine that students take a math test and their
ordered scores look like this.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc

59. Let’s imagine that students take a math test and their
ordered scores look like this. Then let’s imagine they take
a math anxiety survey.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc

60. Let’s imagine that students take a math test and their
ordered scores look like this. Then let’s imagine they take
a math anxiety survey.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.

61. How much do they covary?
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.

62. Bambi has the highest Math Test Score and the highest Math
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Anxiety Score
Math Anxiety Scores
6
5
4
4
3
2
Etc.

63. Belle has the 2nd highest Math Test Score and the 2nd highest
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Score
Math Anxiety Scores
6
5
4
4
3
2
Etc.

64. Billy has the 3rd highest Math Test Score and the 3rd highest Math
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Anxiety Score
Math Anxiety Scores
6
5
4
4
3
2
Etc.

65. Boston has the 4th highest Math Test Score and the 4th highest
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Score
Math Anxiety Scores
6
5
4
4
3
2
Etc.

66. Bryne has the 5th highest Math Test Score and the 5th highest
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Score
Math Anxiety Scores
6
5
4
4
3
2
Etc.

67. Bubba has the 6th highest Math Test Score and the 6th highest
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Score
Math Anxiety Scores
6
5
4
4
3
2
Etc.

68. These two data sets perfectly covary.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.

69. These two data sets perfectly covary.
This means as one changes the other changes.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.

70. These two data sets perfectly covary.
This means as one changes the other changes.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Either in the same
direction

71. These two data sets perfectly covary.
This means as one changes the other changes.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Or opposite
directions
2
3
5
6

72. Covariance is a statistic that describes that relationship.

73. Covariance is a statistic that describes that relationship.
The larger the covariance statistic (either positive or
negative), the more the two samples covary.

74. Covariance is a statistic that describes that relationship.
The larger the covariance statistic (either positive or
negative), the more the two samples covary.
Let’s demonstrate how to calculate covariance by hand.

75. Covariance is a statistic that describes that relationship.
The larger the covariance statistic (either positive or
negative), the more the two samples covary.
Let’s demonstrate how to calculate covariance by hand.
(Although most statistical software can do it for you automatically.)

76. Here is the data set
Student
X i Yi
Test
Scores
Math Anxiety
Scores
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2

77. Here is the sum of each column:
Student
i X i Y
Test
Scores
Math Anxiety
Scores
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24

78. And the mean:
Student
i X i Y
Test
Scores
Math Anxiety
Scores
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

79. Remember – Covariance can only be computed between two or more variables
(e.g., test questions, test scores, ect.) with scores across each variable for each
Student
person.
i X i Y
Test
Scores
Math Anxiety
Scores
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

80. Here is the formula for covariance:
 i X  i Y 
XY
 
X Y
N

  


81. With an explanation for each value:
 i X  i Y 
XY
 
X Y
N

  


82.  i X  i Y 
XY
 
X Y
N

  


83.    
i X i 
Y Anxiety
Scores
XY
X Y
N

  

Test
Scores

84.  i X  i Y 
XY
 
X Y
N

  


85.    
i X i 
Y Each Test
Scores
XY
X Y
N

  


86.  i X  i Y 
XY
 
X Y
N

  


87.    
i X i 
Y Or in this case is
the mean for Test
Scores (82)
XY
X Y
N

  


88.  i X  i Y 
XY
 
X Y
N

  


89.    
i X i 
Y Each
Anxiety
Score
XY
X Y
N

  


90.  i X  i Y 
XY
 
X Y
N

  


91.    
i X i 
Y Or in this case is
the mean for
Anxiety Scores
(24)
XY
X Y
N

  


92. Let the Covariance Calculations Begin!

93. Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

94.  X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

95. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

96. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

97. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

98. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
98

99. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
98 - 82

100. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
16

101. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
16
92 - 82

102. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
16
10
84 - 82

103. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
16
10
2
77 - 82

104. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
16
10
2
-5
73 - 82

105. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
16
10
2
-5
-9
68 - 82

106. Deviation between each
math score and the mean.
 X i   X 
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4
16
10
2
-5
-9
-14

107. i X i Y   i X X  
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4

108. Deviation between each Anxiety
score and its mean.
i X i Y   i X X  
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
  Yi  Y

109. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
5

110. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
5 - 4

111. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
1

112. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
1
6 - 4

113. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
1
2
4 - 4

114. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
1
2
0
4 - 4

115. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
1
2
0
0
3 - 4

116. Deviation between each Anxiety
score and its mean.
i X i Y   i X X     Yi  Y
i Xi Y  i X X   Student
Student
Student
Student
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
1
2
0
0
-1
2 - 4

117. Student
Student
Student
Student
XX i i YY i i  X  X  
     Yi   i i X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4

118. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
product”

119. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16

120. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16 x 1

121. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16

122. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
10 x 2

123. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
2 x 0

124. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
-5 x 0

125. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
-9 x -1

126. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
-14 x -2

127. Student
Student
Student
Student
Multiply the two paired Deviations
to get what is called “the cross
product”
i X i Y   i X X     X i   X Yi Y
i Xi Y  i X X     Yi  Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28

128. Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
XX i i YY i i  X  X  
     Yi   i i X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4

129. Here’s the covariance equation again:
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
XX i i YY i i  X  X  
     Yi   i i X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4

130. Student
Student
Student
Student
   i X i Y
XY
 
i X i Y   i X X    i Y Y     X i   X Yi Y
XX i i YY i i  X  X  
     Yi   i i X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4
X Y
N

  


131.  i X  i Y 
XY
 
X Y
N

  

So far we have calculated a portion
of the numerator of this equation.
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4

132.  i X  i Y 
XY
 
X Y
N

  

Now we will sum or add up the
cross products.
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4

133.  i X  i Y 
XY
 
X Y
N

  

Now we will sum or add up the
cross products.
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4

134.  i X  i Y 
XY
 
X Y
N

  

Now we will sum or add up the
cross products.
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4
Add up

135.  i X  i Y 
XY
 
X Y
N

  

Now we will sum or add up the
cross products.
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
73
Mean 82 4
Add up

136.  i X  i Y 
XY
 
X Y
N

  

Now we will sum or add up the
cross products.
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
73
Mean 82 4

137.  i X  i Y 
XY
 
X Y
N

  

Then we divide the result by the
number of subjects, which in this
case is (6)
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
73
Mean 82 4

138.  i X  i Y 
XY
 
X Y
N

  

Then we divide the result by the
number of subjects, which in this
case is (6)
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4
73 / 6

139.  i X  i Y 
XY
 
X Y
N

  

Then we divide the result by the
number of subjects, which in this
case is (6)
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4
73 / 6 = 12.2

140.  i X  i Y 
XY
 
X Y
12.2
N

  

Covariance = 12.2
Student
Student
Student
Student
i X i Y   i X X    i Y Y     X i   X Yi Y
X X i i Y Y i i  X  X  
     i i Yi   X X Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4
73 / 6 = 12.2

141. Let’s see what the covariance looks like when the direction
of the data goes in the opposite direction:

142. Student
Student
BEFORE
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

143. BEFORE AFTER
X Student
Student
i Y i Student
Student
Test
Scores
Math
Anxiety
Bambi 98 2
Belle 92 3
Billy 84 4
Boston 77 4
Bryne 73 6
Bubba 68 5
Sum 492 24
Mean 82 4
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

144. BEFORE AFTER
X Student
Student
i Y i Student
Student
Test
Scores
Math
Anxiety
Bambi 98 2
Belle 92 3
Billy 84 4
Boston 77 4
Bryne 73 6
Bubba 68 5
Sum 492 24
Mean 82 4
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 5
Belle 92 6
Billy 84 4
Boston 77 4
Bryne 73 3
Bubba 68 2
Sum 492 24
Mean 82 4

145. First we calculate the deviations from the mean:
Student
Student
i X i Y
Test
Scores
Math
Anxiety
Bambi 98 2
Belle 92 3
Billy 84 4
Boston 77 4
Bryne 73 6
Bubba 68 5
Sum 492 24
Mean 82 4

146. First we calculate the deviations from the mean:
Student
Student
i Xi Y  i X X    i Y Y  
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety
Bambi 98 2 16 - 2
Belle 92 3 10 - 1
Billy 84 4 2 0
Boston 77 4 - 5 0
Bryne 73 6 - 9 2
Bubba 68 5 - 14 1
Sum 492 24
Mean 82 4

147. We now compute the Cross Products
Student
Student
i Xi Y  i X X    i Y Y  
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety
x =
x =
x =
x =
x =
x =
Bambi 98 2 16 - 2
Belle 92 3 10 - 1
Billy 84 4 2 0
Boston 77 4 - 5 0
Bryne 73 6 - 9 2
Bubba 68 5 - 14 1
Sum 492 24
Mean 82 4

148. We now compute the Cross Products
Student
Student
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4

149. We sum the cross products and then divide it by the
number of students
Student
Student
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4

150. We sum the cross products and then divide it by the
number of students
Student
Student
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
-73
Mean 82 4

151. We sum the cross products and then divide it by the
number of students
Student
Student
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4
-73 / 6

152. We sum the cross products and then divide it by the
number of students
Student
Student
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4
-73 / 6 = -12.2

153. This is the covariance!
Student
Student
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4
-73 / 6 = -12.2

154. This is the covariance!
Student
Student
Notice that when there is a negative
relationship between two variables
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4
-73 / 6 = -12.2

155. This is the covariance!
Student
Student
Notice that when there is a negative
relationship between two variables
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4
-73 / 6 = -12.2

156. This is the covariance!
Student
Student
Notice that when there is a negative
relationship between two variables
i Xi Y  i X X     i Y Y     X i   X Yi Y
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 2 16 - 2 -32
Belle 92 3 10 - 1 -10
Billy 84 4 2 0 0
Boston 77 4 - 5 0 0
Bryne 73 6 - 9 2 -18
Bubba 68 5 - 14 1 -14
Sum 492 24
Mean 82 4
-73 / 6 = -12.2
The covariance is
negative

157. On the other hand, when the relationship between two
variables is positive . . .

158. On the other hand, when the relationship between two
variables is positive . . .
The covariance will be positive.

159. On the other hand, when the relationship between two
variables is positive . . .
The covariance will be positive.
Student
Student
Student
Student
i X i Y   X i   X   Yi  Y    X i   X Yi Y
XX i i YY i i  X  X        Yi   i i X X
Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4
73 / 6 = 12.2

160. On the other hand, when the relationship between two
variables is positive . . .
The covariance will be positive.
Student
Student
Student
Student
i X i Y   X i   X   Yi  Y    X i   X Yi Y
XX i i YY i i  X  X        Yi   i i X X
Y
Test
Scores
Test
Scores
Math
Anxiety
Math
Anxiety
Test
Scores
Test
Scores
Math
Anxiety
Bambi 98 5 16
Belle 92 6 10
Billy 84 4 2
Boston 77 4 -5
Bryne 73 3 -9
Bubba 68 2 -14
Sum 492 24
Mean 82 4
Bambi 98 5 16 1
Belle 92 6 10 2
Billy 84 4 2 0
Boston 77 4 -5 0
Bryne 73 3 -9 -1
Bubba 68 2 -14 -2
Sum 492 24
Mean 82 4
16
20
0
0
9
28
Student
Student
Test
Scores
Math
Anxiety
Test
Scores
Math
Anxiety Cross Products
Bambi 98 5 16 1 16
Belle 92 6 10 2 20
Billy 84 4 2 0 0
Boston 77 4 -5 0 0
Bryne 73 3 -9 -1 9
Bubba 68 2 -14 -2 28
Sum 492 24
Mean 82 4
73 / 6 = 12.2

161. Why is this important to know?

162. Why is this important to know?
Especially in light of the question we are trying to
answer?

163. Why is this important to know?
Especially in light of the question we are trying to
answer?
The Problem: A pizza café owner wants to know which
type of high school athlete she should market to, by
comparing how many ounces of pizza are consumed
across all three athlete groups. She will control for pizza
preference.

164. Computing covariance helps us-
The Problem: A pizza café owner wants to know which
type of high school athlete she should market to, by
comparing how many ounces of pizza are consumed
across all three athlete groups. She will control for pizza
preference.

165. Computing covariance helps us-
The Problem: A pizza café owner wants to know which
type of high school athlete she should market to, by
comparing how many ounces of pizza are consumed
across all three athlete groups. She will control for pizza
preference.

166. Computing covariance helps us-
The Problem: A pizza café owner wants to know which
type of high school athlete she should market to, by
comparing how many ounces of pizza are consumed
across all three athlete groups. She will control for pizza
preference.
Meaning that we want to know what the difference
between the three groups would be if we took away all
of the covariance between pizza preference and amount
of ounces of pizza eaten.

167. Computing covariance helps us-
The Problem: A pizza café owner wants to know which
type of high school athlete she should market to, by
comparing how many ounces of pizza are consumed
across all three athlete groups. She will control for pizza
preference.
If we did not take out the covariance, then a bunch of
soccer players may like pizza not because they are soccer
players (our research question) but because they just
LOVE PIZZA (not our research question).

168. Computing covariance helps us-
The Problem: A pizza café owner wants to know which
type of high school athlete she should market to, by
comparing how many ounces of pizza are consumed
across all three athlete groups. She will control for pizza
preference.
Because their love of pizza is not what we are testing, we
will control for it by computing covariance and see how
much of the fact that they are soccer players really
affects the amount of ounces of pizza they eat.

169. So, let’s begin by running a One-way ANOVA without
removing the covariance between pizza preference and
ounces eaten by athlete type.

170. Here’s the data
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

171. Here are the results of the one-way ANOVA for this
data set:
Sums of
Squares df Mean Square F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups
(error) 1587.4 33 48.1
Total 1626.3
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

172. Here are the results of the one-way ANOVA for this
data set:
Sums of
Squares df Mean Square F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups
(error) 1587.4 33 48.1
Total 1626.3
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
As you will recall, an F-ratio
1 or lower with any ANOVA
method is not significant.

173. Here are the results of the one-way ANOVA for this
data set:
Sums of
Squares df Mean Square F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups
(error) 1587.4 33 48.1
Total 1626.3
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
As you will recall, an F-ratio
1 or lower with any ANOVA
method is not significant.

174. Then
After calculating the covariance between pizza
preference and ounces of pizza eaten in one
sitting, we find that there is a positive
relationship.

175. After calculating the covariance between pizza preference
and ounces of pizza eaten in one sitting, we find that there
is a positive relationship.
Pizza Preference (scale 1-10)
Football Basketball Soccer
7.0 3.0 7.5
5.0 8.0 4.5
3.5 4.5 3.5
9.0 9.5 6.0
7.0 6.5 6.0
8.0 7.0 4.5
6.5 7.5 6.0
7.5 9.0 1.5
2.5 8.5 6.5
9.0 4.0 5.0
8.0 7.5 5.5
5.0 8.0 4.0
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

176. After calculating the covariance between pizza preference
and ounces of pizza eaten in one sitting, we find that there
is a positive relationship.
Pizza Preference (scale 1-10)
Football Basketball Soccer
7.0 3.0 7.5
5.0 8.0 4.5
3.5 4.5 3.5
9.0 9.5 6.0
7.0 6.5 6.0
8.0 7.0 4.5
6.5 7.5 6.0
7.5 9.0 1.5
2.5 8.5 6.5
9.0 4.0 5.0
8.0 7.5 5.5
5.0 8.0 4.0
Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
Covariance =
12.1

177. After running the Analysis of Covariance on the data and
partialling out pizza reference, here is the resulting ANOVA
table:

178. After running the Analysis of Covariance on the data and
partialling out pizza reference, here is the resulting ANOVA
table:
SS df MS F
Adjusted means (BG) 74.5 2 37.2 3.8
Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34

179. After running the Analysis of Covariance on the data and
partialling out pizza reference, here is the resulting ANOVA
table:
Adjusted means – after we took out
the covariance between the two
variables: Type of Athlete and Pizza
Preference (the covariate)
SS df MS F
Adjusted means (BG) 74.5 2 37.2 3.8
Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34

180. After running the Analysis of Covariance on the data and
partialling out pizza reference, here is the resulting ANOVA
table:
SS df MS F
Adjusted means (BG) 74.5 2 37.2 3.8
Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
Notice the F-ratio is larger
making it more likely to be
significant.

181. Let’s compare the F-ratio for just the ANOVA

182. Let’s compare the F-ratio for just the ANOVA
Before
SS df MS F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups (error) 1587.4 33 48.1
Total 1626.3

183. Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
Before
SS df MS F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups (error) 1587.4 33 48.1
Total 1626.3

184. Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
Before
SS df MS F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups (error) 1587.4 33 48.1
Total 1626.3
After
SS df MS F-Ratio
Adjusted means (BG) 74.5 2 37.2 3.8
Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34

185. Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
Before
SS df MS F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups (error) 1587.4 33 48.1
Total 1626.3
After
SS df MS F-Ratio
Adjusted means (BG) 74.5 2 37.2 3.8
Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
So we would conclude that there is a significant difference between football,
basketball & soccer players in terms of the ounces of pizza they eat, that is,
when we control for pizza preference.

186. Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
Before
SS df MS F-Ratio
Between Groups 38.9 2 19.4 0.4
Within Groups (error) 1587.4 33 48.1
Total 1626.3
After
SS df MS F-Ratio
Adjusted means (BG) 74.5 2 37.2 3.8
Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
So we would conclude that there is a significant difference between football,
basketball & soccer players in terms of the ounces of pizza they eat, that is,
when we control for pizza preference.

187. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.

188. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8

189. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3

190. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Notice that after controlling for
pizza preference, the mean for
Basketball players drops

191. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Notice that after controlling for
pizza preference, the mean for
Basketball players drops

192. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Notice that after controlling for
pizza preference, the mean for
Basketball players drops

193. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
And the mean for
Soccer players
INCREASES!

194. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
And the mean for
Soccer players
INCREASES!

195. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
And the mean for
Soccer players
INCREASES!

196. We can even adjust the original means for amount of
ounces of pizza eaten, after controlling for preference.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
That’s the Power of
ANCOVA!

197. Important note,
The more the covariate (pizza preference) covaries with
the independent variable (type of athlete) . .

198. Important note,
The more the covariate (pizza preference) covaries with
the independent variable (type of athlete) . . . the
bigger the adjustment will be between original and
adjusted means.

199. Important note,
The more the covariate (pizza preference) covaries with
the independent variable (type of athlete) . . . the
bigger the adjustment will be between original and
adjusted means.
Meaning they share a larger covariance
value (either positive or negative).

200. Important note,
The more the covariate (pizza preference) covaries with
the independent variable (type of athlete) . . . the
bigger the adjustment will be between original and
adjusted means.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3

201. Important note,
The more the covariate (pizza preference) covaries with
the independent variable (type of athlete) . . . the
bigger the adjustment will be between original and
adjusted means.
Original Means
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Adjusted Means
Big Adjustments
(after controlling for the covariance)
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3

202. One final note

203. In this case the covariate (the thing we were controlling
for) was a continuous variable like
• Ounces of pizza eaten
• Time it takes to eat pizza
• The weight of each athlete.

204. In this case the covariate (the thing we were controlling
for) was a continuous variable like
• Ounces of pizza eaten
• Time it takes to eat pizza
• The weight of each athlete.

205. In this case the covariate (the thing we were controlling
for) was a continuous variable like
• Ounces of pizza eaten
• Time it takes to eat pizza
• The weight of each athlete.
But it also can be categorical (one or the other)

206. In this case the covariate (the thing we were controlling
for) was a continuous variable like
• Ounces of pizza eaten
• Time it takes to eat pizza
• The weight of each athlete.
But it also can be categorical (one or the other)
• Year in School (Sophomores, Juniors, or Seniors)
• Gender (Male or Female)
• Religious Affiliation (Muslim, Catholic, etc.)

207. In summary

208. Analysis of Covariance is a powerful tool that
makes it possible to control for any variable that is
not of interest (eg. pizza preference)

209. Analysis of Covariance is a powerful tool that
makes it possible to control for any variable that is
not of interest (eg. pizza preference) in order to
see the true effect of the variable of interest (type
of athlete) on a dependent variable of interest
(ounces of pizza eaten)

210. There are more complex methods such as
Factorial ANCOVA, Repeated measures ANCOVA
and Multivariate ANCOVA.

211. There are more complex methods such as
Factorial ANCOVA, Repeated measures ANCOVA
and Multivariate ANCOVA.
This presentation gives you the conceptual
foundation necessary to understand the Analysis
of Covariance elements of these methods.